Problem: What is the slope of the line tangent to $f(x) = 2x^{2}-3x-5$ at $x = 0$ ?
The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(2(x+\Delta x)^{2}-3(x+\Delta x)-5) - (2x^{2}-3x-5)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(2(x^{2}+2x \Delta x+\Delta x^{2})-3(x+\Delta x)-5) - (2x^{2}-3x-5)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{2x^{2}+4(x \Delta x)+2\Delta x^{2}-3x-3(\Delta x)-5-2x^{2}+3x+5}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{4(x \Delta x)+2\Delta x^{2}-3(\Delta x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} 4x+2(\Delta x)-3$ $ = 4x-3$ $ = (4)(0)-3$ $ = -3$